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We consider the effects connected with the detailed radiative transfer duringthe epoch of cosmological recombination on the ionization history of ourUniverse. We focus on the escape of photons from the hydrogen Lyman-alpharesonance at redshifts 600<~ z <~ 2000, one of two key mechanisms defining therate of cosmological recombination. We approach this problem within thestandard formulation, and corrections due to two-photon interactions aredeferred to another paper. As a main result we show here that within anon-stationary approach to the escape problem, the resulting correction in thefree electron fraction, N_e, is about ~1.6-1.8% in the redshift range800<~z<~1200. Therefore the discussed process results in one of the largestmodifications to the ionization history close to the maximum ofThomson-visibility function at z~1100 considered so far. We prove our resultsboth numerically and analytically, deriving the escape probability, andconsidering both Lyman-alpha line emission and line absorption in a waydifferent from the Sobolev approximation. In particular, we give a detailedderivation of the Sobolev escape probability during hydrogen recombination, andexplain the underlying assumptions. We then discuss the escape of photons forthe case of coherent scattering in the lab frame, solving this problemanalytically in the quasi-stationary approximation and also in thetime-dependent case. We show here that during hydrogen recombination theSobolev approximation for the escape probability is not valid at the level ofDP/P~5-10%. This is because during recombination the ionization degree changessignificantly over a characteristic time Dz/z~10%, so that at percent levelaccuracy the photon distribution is not evolving (abridged)

Time-Dependent Corrections to the Ly-alpha Escape Probability During Cosmological Hydrogen Recombination